# Modeling and Analysis of Torsional Stiffness in Rehabilitation Robot Joints Using Fractal Theory

## TL;DR

This paper uses fractal theory to model and analyze torsional stiffness in rehabilitation robot joints, improving design and performance.

## Contribution

The novel contribution is applying fractal theory to model surface roughness effects on torsional stiffness in traction drive joints.

## Key findings

- Contact stiffness increases with normal load, contact length, and radius, but decreases with tangential load and roughness parameter.
- Stiffness has a non-monotonic relationship with fractal dimension, peaking at intermediate values.
- Overall system stiffness shows similar parameter dependencies with slight decreases under increasing output load with sufficient preload.

## Abstract

The torsional stiffness of rehabilitation robot joints is a critical performance determinant, significantly affecting motion accuracy, stability, and user comfort. This paper introduces an innovative traction drive mechanism that transmits torque through friction forces, overcoming mechanical impact issues of traditional gear transmissions, though accurately modeling surface roughness effects remains challenging. Based on fractal theory, this study presents a comprehensive torsional stiffness analysis for advanced traction drive joints. Surface topography is characterized using the Weierstrass–Mandelbrot function, and a contact mechanics model accounting for elastic–plastic deformation of micro-asperities is developed to derive the tangential stiffness of individual contact pairs. Static force analysis determines load distribution, and overall joint torsional stiffness is calculated through the integration of individual contact contributions. Parametric analyses reveal that contact stiffness increases with normal load, contact length, and radius, while decreasing with the tangential load and roughness parameter. Stiffness exhibits a non-monotonic relationship with fractal dimension, reaching a maximum at intermediate values. Overall system stiffness demonstrates similar parameter dependencies, with a slight decrease under increasing output load when sufficient preload is applied. This fractal-based model enables more accurate stiffness prediction and offers valuable theoretical guidance for design optimization and performance improvement in rehabilitation robot joints.